The kinetic energy increases because the energy levels depend on the volume the electrons are allowed to occupy. A smaller volume (or more electrons in the same volume) will increase the total energy of the system. For some reason, we like to think of a force as arising from a potential energy, but we have no problem with a pressure being due to the kinetic energy of a gas. The exclusion principle is just the pressure due to the kinetic energy of the electrons–which increases when they are compressed. (It’s a wave-function thing, not dependent on temperature.)

-Dave

http://www.particlezoo.net/
http://www.youtube.com/watch?v=j50ZssEojtM
… and especially: http://www.particleadventure.org/

You can use density matrices in something like 5 very much different ways. One of them is as you’ve described. But you can also take the pure states that are used in state vectors and convert them into “pure density matrices”. These are exactly equivalent to the usual state vectors in terms of information content, except that state vectors have arbitrary complex phases that have no physical interpretation. (The physical uses of complex phase are shared with between density matrices, otherwise they wouldn’t work.)

“So I wonder what physical significance there would have to be to a DM formulation, considering it is “a way of looking at things” more than an actual situation of thing.”

If you’re interested in the actual thing (or ontology), rather than mathematical ways of describing activity, then you’re probably interested in Bohmian mechanics. I’ve linked to a paper that discusses density matrices with respect to Bohmian mechanics which was the first reference I saw on arXiv in 30 seconds, but there are a lot of more complete articles. It’s an active field.

From the density matrix point of view, state vectors are a kluge that is done so that a quantum state can be converted from its natural bilinear condition, to an unnatural, but mathematically convenient linear one. The mathematical convenience is “linear superposition”, which is a principle that allows you to describe a large number of quantum states from taking linear combinations of other states, for example, the usual basis states in spin-1/2 of spin up and spin down.

Linear superposition (as opposed to interference) does not translate into a literal experiment that can be performed in the lab. That would be like taking, for example, combining a spin up electron with a spin down electron and expecting to get an electron oriented with spin in the x direction. You will not get this as a result. Instead, you will get two electrons and to model them you will require a more complicated state vector, i.e. ++, +-, -+, — as basis. Of course interference works fine in density matrix calculations just like with state vectors.

The bilinearity of density matrices can be thought of as describing a quantum state in terms of how it acts as an operator. The pure density matrices are projection operators. This makes density matrices, rather than state vectors, the natural way to define the “Consistent Histories” interpretation of quantum mechanics (which generalizes QM to quantum cosmology). Read the article on consistent histories at Wikipedia, or do a search for arXiv articles.

So density matrices have an interpretation of quantum mechanics that is devoted to their use and avoids the state vector formalism. At the very least this makes density matrices the equal of state vectors but for those who prefer quantum mechanics that naturally fits into quantum cosmology, density matrices are superior. And their sign does not change when you swap particles in them.

Mathematically, this comes about because a density matrix is made from copies of the same state vector, a bra and a ket. When you swap two particles, you get a minus sign for the bra and a minus sign for the ket. Since (-1)(-1) = +1, the density matrix is unchanged.

This is true only for the state vector formalism of quantum mechanics. In density matrix formalism, the arbitrary complex phases are eliminated and exchanging two particles leaves one with the same density matrix state.

Since density matrices and state vectors give identical results in quantum mechanics, it is not possible to say which one is fundamental. Consequently, it cannot be said that swapping fermions changes a state.

From the density matrix point of view, a state vector is a sort of square root of a density matrix. It is not surprising that one must supply a phase (in analogy with the sign that arises with a square root). But if the fundamental nature is the density matrix (which can represent states that state vectors cannot and therefore are more general), then the minus sign that shows up when fermions are swapped is just an accident of mathematics, not a fundamental part of the physics. Instead, the difference between fermions and bosons is that the occupation numbers for fermions can only be 0 or 1 while bosons can count higher.

Google “nine formulations of quantum mechanics” for more info.

One Loop Vacuum Polarization without Infinities (hep-th/0207069)

A technique for avoiding infinite integrals in the calculation of the one-loop diagram contribution to the vacuum polarization component of an atomic energy level is presented. This makes renormalization unnecessary. Infinite integrals do not occur because, as it is shown, no delta functions are required for the Green’s functions. Thus there are none to overlap. This procedure is shown to produce the same formula as the one obtained by dimensional renormalization.

(But I digress…)